Theorem disji 5133

Description: Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑋 = 𝑌. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
disji.1	⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
disji.2	⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)

Assertion

Ref	Expression
disji	⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝑍 ∈ 𝐶 ∧ 𝑍 ∈ 𝐷)) → 𝑋 = 𝑌)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑋   𝑥,𝑌

Allowed substitution hints:   𝐵(𝑥)   𝑍(𝑥)

Proof of Theorem disji

Step	Hyp	Ref	Expression
1		inelcm 4471	. 2 ⊢ ((𝑍 ∈ 𝐶 ∧ 𝑍 ∈ 𝐷) → (𝐶 ∩ 𝐷) ≠ ∅)
2		disji.1	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
3		disji.2	. . . . . 6 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
4	2, 3	disji2 5132	. . . . 5 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ 𝑋 ≠ 𝑌) → (𝐶 ∩ 𝐷) = ∅)
5	4	3expia 1120	. . . 4 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋 ≠ 𝑌 → (𝐶 ∩ 𝐷) = ∅))
6	5	necon1d 2960	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐶 ∩ 𝐷) ≠ ∅ → 𝑋 = 𝑌))
7	6	3impia 1116	. 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐶 ∩ 𝐷) ≠ ∅) → 𝑋 = 𝑌)
8	1, 7	syl3an3 1164	1 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝑍 ∈ 𝐶 ∧ 𝑍 ∈ 𝐷)) → 𝑋 = 𝑌)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   ∩ cin 3962  ∅c0 4339  Disj wdisj 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rmo 3378  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-in 3970  df-nul 4340  df-disj 5116

This theorem is referenced by:  volfiniun  25596  fnpreimac  32688